Numerical

1
Let $r$ be the radius of the circle, which touches $x$ - axis at point $(a, 0), a<0$ and the parabola $\mathrm{y}^2=9 x$ at the point $(4,6)$. Then $r$ is equal to ______.
JEE Main 2025 (Online) 8th April Evening Shift
2

Let $y^2=12 x$ be the parabola and $S$ be its focus. Let $P Q$ be a focal chord of the parabola such that $(S P)(S Q)=\frac{147}{4}$. Let $C$ be the circle described taking $P Q$ as a diameter. If the equation of a circle $C$ is $64 x^2+64 y^2-\alpha x-64 \sqrt{3} y=\beta$, then $\beta-\alpha$ is equal to $\qquad$ .

JEE Main 2025 (Online) 29th January Evening Shift
3

Let $A$ and $B$ be the two points of intersection of the line $y+5=0$ and the mirror image of the parabola $y^2=4 x$ with respect to the line $x+y+4=0$. If $d$ denotes the distance between $A$ and $B$, and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y^2=4 x$, then the value of $(a+d)$ is __________.

JEE Main 2025 (Online) 28th January Evening Shift
4

The focus of the parabola $y^2=4 x+16$ is the centre of the circle $C$ of radius 5 . If the values of $\lambda$, for which C passes through the point of intersection of the lines $3 x-y=0$ and $x+\lambda y=4$, are $\lambda_1$ and $\lambda_2, \lambda_1<\lambda_2$, then $12 \lambda_1+29 \lambda_2$ is equal to ________ .

JEE Main 2025 (Online) 23rd January Evening Shift
5

Let $$A, B$$ and $$C$$ be three points on the parabola $$y^2=6 x$$ and let the line segment $$A B$$ meet the line $$L$$ through $$C$$ parallel to the $$x$$-axis at the point $$D$$. Let $$M$$ and $$N$$ respectively be the feet of the perpendiculars from $$A$$ and $$B$$ on $$L$$. Then $$\left(\frac{A M \cdot B N}{C D}\right)^2$$ is equal to __________.

JEE Main 2024 (Online) 9th April Evening Shift
6

Consider the circle $$C: x^2+y^2=4$$ and the parabola $$P: y^2=8 x$$. If the set of all values of $$\alpha$$, for which three chords of the circle $$C$$ on three distinct lines passing through the point $$(\alpha, 0)$$ are bisected by the parabola $$P$$ is the interval $$(p, q)$$, then $$(2 q-p)^2$$ is equal to __________.

JEE Main 2024 (Online) 9th April Evening Shift
7

Let a conic $$C$$ pass through the point $$(4,-2)$$ and $$P(x, y), x \geq 3$$, be any point on $$C$$. Let the slope of the line touching the conic $$C$$ only at a single point $$P$$ be half the slope of the line joining the points $$P$$ and $$(3,-5)$$. If the focal distance of the point $$(7,1)$$ on $$C$$ is $$d$$, then $$12 d$$ equals ________.

JEE Main 2024 (Online) 6th April Morning Shift
8

Let $$L_1, L_2$$ be the lines passing through the point $$P(0,1)$$ and touching the parabola $$9 x^2+12 x+18 y-14=0$$. Let $$Q$$ and $$R$$ be the points on the lines $$L_1$$ and $$L_2$$ such that the $$\triangle P Q R$$ is an isosceles triangle with base $$Q R$$. If the slopes of the lines $$Q R$$ are $$m_1$$ and $$m_2$$, then $$16\left(m_1^2+m_2^2\right)$$ is equal to __________.

JEE Main 2024 (Online) 6th April Morning Shift
9

Let a line perpendicular to the line $$2 x-y=10$$ touch the parabola $$y^2=4(x-9)$$ at the point P. The distance of the point P from the centre of the circle $$x^2+y^2-14 x-8 y+56=0$$ is __________.

JEE Main 2024 (Online) 5th April Evening Shift
10

Suppose $$\mathrm{AB}$$ is a focal chord of the parabola $$y^2=12 x$$ of length $$l$$ and slope $$\mathrm{m}<\sqrt{3}$$. If the distance of the chord $$\mathrm{AB}$$ from the origin is $$\mathrm{d}$$, then $$l \mathrm{~d}^2$$ is equal to _________.

JEE Main 2024 (Online) 5th April Morning Shift
11

Let the length of the focal chord PQ of the parabola $$y^2=12 x$$ be 15 units. If the distance of $$\mathrm{PQ}$$ from the origin is $$\mathrm{p}$$, then $$10 \mathrm{p}^2$$ is equal to __________.

JEE Main 2024 (Online) 4th April Morning Shift
12
Let the line $\mathrm{L}: \sqrt{2} x+y=\alpha$ pass through the point of the intersection $\mathrm{P}$ (in the first quadrant) of the circle $x^2+y^2=3$ and the parabola $x^2=2 y$. Let the line $\mathrm{L}$ touch two circles $\mathrm{C}_1$ and $\mathrm{C}_2$ of equal radius $2 \sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the $y$-axis, then the square of the area of the triangle $\mathrm{PQ}_1 \mathrm{Q}_2$ is equal to ___________.
JEE Main 2024 (Online) 1st February Morning Shift
13

Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2=4 x$$. If $$P$$ also lies on the chord of the parabola $$x^2=8 y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha-28)(\beta-8)$$ is equal to _________.

JEE Main 2024 (Online) 29th January Evening Shift
14

Let the tangent to the parabola $$\mathrm{y}^{2}=12 \mathrm{x}$$ at the point $$(3, \alpha)$$ be perpendicular to the line $$2 x+2 y=3$$. Then the square of distance of the point $$(6,-4)$$ from the normal to the hyperbola $$\alpha^{2} x^{2}-9 y^{2}=9 \alpha^{2}$$ at its point $$(\alpha-1, \alpha+2)$$ is equal to _________.

JEE Main 2023 (Online) 11th April Evening Shift
15

Let a common tangent to the curves $${y^2} = 4x$$ and $${(x - 4)^2} + {y^2} = 16$$ touch the curves at the points P and Q. Then $${(PQ)^2}$$ is equal to __________.

JEE Main 2023 (Online) 10th April Morning Shift
16

The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R}(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$\mathrm{P}$$ and $$\mathrm{Q}$$, then $$\frac{\beta^{2}}{\alpha}$$ is equal to _______________.

JEE Main 2023 (Online) 8th April Evening Shift
17

Let the tangent to the curve $$x^{2}+2 x-4 y+9=0$$ at the point $$\mathrm{P}(1,3)$$ on it meet the $$y$$-axis at $$\mathrm{A}$$. Let the line passing through $$\mathrm{P}$$ and parallel to the line $$x-3 y=6$$ meet the parabola $$y^{2}=4 x$$ at $$\mathrm{B}$$. If $$\mathrm{B}$$ lies on the line $$2 x-3 y=8$$, then $$(\mathrm{AB})^{2}$$ is equal to ___________.

JEE Main 2023 (Online) 6th April Morning Shift
18

If the $$x$$-intercept of a focal chord of the parabola $$y^{2}=8x+4y+4$$ is 3, then the length of this chord is equal to ____________.

JEE Main 2023 (Online) 1st February Evening Shift
19
Let $\mathrm{S}$ be the set of all $\mathrm{a} \in \mathrm{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(\mathrm{b}$, c), b, c $\in \mathbb{N}$, on the parabola $y^{2}=2 \mathrm{a} x$ and the lines $x=\mathrm{b}, y=0$ is $16 $ unit2, then $\sum\limits_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to :
JEE Main 2023 (Online) 31st January Evening Shift
20

A triangle is formed by the tangents at the point (2, 2) on the curves $$y^2=2x$$ and $$x^2+y^2=4x$$, and the line $$x+y+2=0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to ___________.

JEE Main 2023 (Online) 29th January Evening Shift
21

Two tangent lines $$l_{1}$$ and $$l_{2}$$ are drawn from the point $$(2,0)$$ to the parabola $$2 \mathrm{y}^{2}=-x$$. If the lines $$l_{1}$$ and $$l_{2}$$ are also tangent to the circle $$(x-5)^{2}+y^{2}=r$$, then 17r is equal to ___________.

JEE Main 2022 (Online) 28th July Evening Shift
22

The sum of diameters of the circles that touch (i) the parabola $$75 x^{2}=64(5 y-3)$$ at the point $$\left(\frac{8}{5}, \frac{6}{5}\right)$$ and (ii) the $$y$$-axis, is equal to ______________.

JEE Main 2022 (Online) 25th July Morning Shift
23

Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $$\Delta$$POQ is equal to ___________.

JEE Main 2022 (Online) 30th June Morning Shift
24

A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola $$y = {\left( {x - {1 \over 4}} \right)^2} + \alpha $$, where $$\alpha$$ > 0. Then (4$$\alpha$$ $$-$$ 8)2 is equal to ______________.

JEE Main 2022 (Online) 27th June Morning Shift
25

Let the common tangents to the curves $$4({x^2} + {y^2}) = 9$$ and $${y^2} = 4x$$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $${l \over {{e^2}}}$$ is equal to ______________.

JEE Main 2022 (Online) 26th June Morning Shift
26

Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = ____________.

JEE Main 2022 (Online) 24th June Evening Shift
27
A tangent line L is drawn at the point (2, $$-$$4) on the parabola y2 = 8x. If the line L is also tangent to the circle x2 + y2 = a, then 'a' is equal to ___________.
JEE Main 2021 (Online) 31st August Evening Shift
28
If the point on the curve y2 = 6x, nearest to the point $$\left( {3,{3 \over 2}} \right)$$ is ($$\alpha$$, $$\beta$$), then 2($$\alpha$$ + $$\beta$$) is equal to _____________.
JEE Main 2021 (Online) 20th July Evening Shift
29
Let y = mx + c, m > 0 be the focal chord of y2 = $$-$$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$$\sqrt 2 $$ (m + c) is equal to _____________.
JEE Main 2021 (Online) 20th July Morning Shift
30
A line is a common tangent to the circle (x $$-$$ 3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to _________.
JEE Main 2021 (Online) 25th February Evening Shift
31
If the tangent to the curve, y = ex at a point (c, ec) and the normal to the parabola, y2 = 4x at the point (1, 2) intersect at the same point on the x-axis, then the value of c is ________ .
JEE Main 2020 (Online) 3rd September Evening Slot
32
Let a line y = mx (m > 0) intersect the parabola, y2 = x at a point P, other than the origin. Let the tangent to it at P meet the x-axis at the point Q. If area ($$\Delta $$OPQ) = 4 sq. units, then m is equal to __________.
JEE Main 2020 (Online) 8th January Evening Slot

MCQ (Single Correct Answer)

1

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P, whose abscissa is $-$2, is

JEE Main 2025 (Online) 7th April Morning Shift
2

A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}: y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $A B$, parabola $P$ and the $x$-axis, is :

JEE Main 2025 (Online) 4th April Evening Shift
3

The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2 \sqrt{2}$ units from the origin, respectively. If the point $(1, k)$ lies on the parabola, then a possible value of k is :

JEE Main 2025 (Online) 4th April Evening Shift
4
The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is
JEE Main 2025 (Online) 3rd April Morning Shift
5
Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the chord $P Q$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$, then $m^2+n^2$ is equal to :
JEE Main 2025 (Online) 2nd April Evening Shift
6

Let the focal chord PQ of the parabola $y^2=4 x$ make an angle of $60^{\circ}$ with the positive $x$ axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the $y$-axis at the point $(0, \alpha)$, then $5 \alpha^2$ is equal to:

JEE Main 2025 (Online) 2nd April Morning Shift
7

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

JEE Main 2025 (Online) 29th January Morning Shift
8

Let ABCD be a trapezium whose vertices lie on the parabola $\mathrm{y}^2=4 \mathrm{x}$. Let the sides AD and BC of the trapezium be parallel to $y$-axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of $A B C D$ is

JEE Main 2025 (Online) 28th January Morning Shift
9

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is $\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0$, then $\alpha+\beta+\gamma$ is equal to :

JEE Main 2025 (Online) 24th January Evening Shift
10

Let the shortest distance from $(a, 0), a>0$, to the parabola $y^2=4 x$ be 4 . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :

JEE Main 2025 (Online) 23rd January Evening Shift
11

If the line $3 x-2 y+12=0$ intersects the parabola $4 y=3 x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment AB subtends an angle equal to

JEE Main 2025 (Online) 23rd January Morning Shift
12

Let $\mathrm{P}(4,4 \sqrt{3})$ be a point on the parabola $y^2=4 \mathrm{a} x$ and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

JEE Main 2025 (Online) 22nd January Evening Shift
13

Let the parabola $y=x^2+\mathrm{p} x-3$, meet the coordinate axes at the points $\mathrm{P}, \mathrm{Q}$ and R . If the circle C with centre at $(-1,-1)$ passes through the points $P, Q$ and $R$, then the area of $\triangle P Q R$ is :

JEE Main 2025 (Online) 22nd January Morning Shift
14

Let $$C$$ be the circle of minimum area touching the parabola $$y=6-x^2$$ and the lines $$y=\sqrt{3}|x|$$. Then, which one of the following points lies on the circle $$C$$ ?

JEE Main 2024 (Online) 6th April Morning Shift
15

Let $$P Q$$ be a chord of the parabola $$y^2=12 x$$ and the midpoint of $$P Q$$ be at $$(4,1)$$. Then, which of the following point lies on the line passing through the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ ?

JEE Main 2024 (Online) 4th April Evening Shift
16
If the shortest distance of the parabola $y^2=4 x$ from the centre of the circle $x^2+y^2-4 x-16 y+64=0$ is $\mathrm{d}$, then $\mathrm{d}^2$ is equal to :
JEE Main 2024 (Online) 27th January Morning Shift
17

Let $$\mathrm{PQ}$$ be a focal chord of the parabola $$y^{2}=36 x$$ of length 100 , making an acute angle with the positive $$x$$-axis. Let the ordinate of $$\mathrm{P}$$ be positive and $$\mathrm{M}$$ be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $$\mathrm{PQ}$$?

JEE Main 2023 (Online) 13th April Morning Shift
18

Let $$\mathrm{A}(0,1), \mathrm{B}(1,1)$$ and $$\mathrm{C}(1,0)$$ be the mid-points of the sides of a triangle with incentre at the point $$\mathrm{D}$$. If the focus of the parabola $$y^{2}=4 \mathrm{ax}$$ passing through $$\mathrm{D}$$ is $$(\alpha+\beta \sqrt{2}, 0)$$, where $$\alpha$$ and $$\beta$$ are rational numbers, then $$\frac{\alpha}{\beta^{2}}$$ is equal to :

JEE Main 2023 (Online) 8th April Evening Shift
19

Let $$R$$ be the focus of the parabola $$y^{2}=20 x$$ and the line $$y=m x+c$$ intersect the parabola at two points $$P$$ and $$Q$$.

Let the point $$G(10,10)$$ be the centroid of the triangle $$P Q R$$. If $$c-m=6$$, then $$(P Q)^{2}$$ is :

JEE Main 2023 (Online) 8th April Morning Shift
20

Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then

$$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$$ :

JEE Main 2023 (Online) 31st January Morning Shift
21
Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(Q R)^2$ is equal to :
JEE Main 2023 (Online) 30th January Evening Shift
22
The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :
JEE Main 2023 (Online) 30th January Evening Shift
23

If $$\mathrm{P}(\mathrm{h}, \mathrm{k})$$ be a point on the parabola $$x=4 y^{2}$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the distance of $$\mathrm{P}$$ from the directrix of the parabola $$\quad y^{2}=4(x+y)$$ is equal to :

JEE Main 2023 (Online) 30th January Morning Shift
24

If the tangent at a point P on the parabola $$y^2=3x$$ is parallel to the line $$x+2y=1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4}+\frac{y^2}{1}=1$$ are perpendicular to the line $$x-y=2$$, then the area of the triangle PQR is :

JEE Main 2023 (Online) 29th January Evening Shift
25

The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of its circumcentre is :

JEE Main 2023 (Online) 25th January Evening Shift
26

The distance of the point $$(6,-2\sqrt2)$$ from the common tangent $$\mathrm{y=mx+c,m > 0}$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :

JEE Main 2023 (Online) 25th January Morning Shift
27

The equations of the sides AB and AC of a triangle ABC are $$(\lambda+1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda)y+\lambda=0$$ respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola $$y^2=6x$$ in the first quadrant is :

JEE Main 2023 (Online) 24th January Evening Shift
28

Let a tangent to the curve $$\mathrm{y^2=24x}$$ meet the curve $$xy = 2$$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

JEE Main 2023 (Online) 24th January Morning Shift
29

Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$\mathrm{H}: x^{2}-y^{2}=4$$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :

JEE Main 2022 (Online) 29th July Morning Shift
30

If the tangents drawn at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the parabola $$y^{2}=2 x-3$$ intersect at the point $$R(0,1)$$, then the orthocentre of the triangle $$P Q R$$ is :

JEE Main 2022 (Online) 28th July Morning Shift
31

If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x+y=a$$, is 16, then $$|a|$$ is equal to :

JEE Main 2022 (Online) 27th July Evening Shift
32

Let $$P(a, b)$$ be a point on the parabola $$y^{2}=8 x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^{2}+y^{2}-10 x-14 y+65=0$$. Let $$A$$ be the product of all possible values of $$a$$ and $$B$$ be the product of all possible values of $$b$$. Then the value of $$A+B$$ is equal to :

JEE Main 2022 (Online) 27th July Morning Shift
33

Let $$\mathrm{P}$$ and $$\mathrm{Q}$$ be any points on the curves $$(x-1)^{2}+(y+1)^{2}=1$$ and $$y=x^{2}$$, respectively. The distance between $$P$$ and $$Q$$ is minimum for some value of the abscissa of $$P$$ in the interval :

JEE Main 2022 (Online) 26th July Evening Shift
34

The equation of a common tangent to the parabolas $$y=x^{2}$$ and $$y=-(x-2)^{2}$$ is

JEE Main 2022 (Online) 26th July Evening Shift
35

The tangents at the points $$A(1,3)$$ and $$B(1,-1)$$ on the parabola $$y^{2}-2 x-2 y=1$$ meet at the point $$P$$. Then the area (in unit $${ }^{2}$$ ) of the triangle $$P A B$$ is :

JEE Main 2022 (Online) 25th July Evening Shift
36

Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of $${\pi \over 4}$$ with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is :

JEE Main 2022 (Online) 29th June Evening Shift
37

Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of $${\pi \over 2}$$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $${a^2} > {b^2}$$. If e is the eccentricity of the ellipse E, then the value of $${1 \over {{e^2}}}$$ is equal to :

JEE Main 2022 (Online) 29th June Morning Shift
38

If vertex of a parabola is (2, $$-$$1) and the equation of its directrix is 4x $$-$$ 3y = 21, then the length of its latus rectum is :

JEE Main 2022 (Online) 28th June Evening Shift
39

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $$3x + y - 29 = 0$$, is $${x^2} + a{y^2} + bxy + cx + dy + k = 0$$, then $$a + b + c + d + k$$ is equal to :

JEE Main 2022 (Online) 27th June Evening Shift
40

Let the normal at the point on the parabola y2 = 6x pass through the point (5, $$-$$8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

JEE Main 2022 (Online) 26th June Morning Shift
41

If the line $$y = 4 + kx,\,k > 0$$, is the tangent to the parabola $$y = x - {x^2}$$ at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

JEE Main 2022 (Online) 25th June Evening Shift
42

If $$y = {m_1}x + {c_1}$$ and $$y = {m_2}x + {c_2}$$, $${m_1} \ne {m_2}$$ are two common tangents of circle $${x^2} + {y^2} = 2$$ and parabola y2 = x, then the value of $$8|{m_1}{m_2}|$$ is equal to :

JEE Main 2022 (Online) 25th June Morning Shift
43

Let $$x = 2t$$, $$y = {{{t^2}} \over 3}$$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $$SA \bot BA$$, where A is any point on the conic. If k is the ordinate of the centroid of the $$\Delta$$SAB, then $$\mathop {\lim }\limits_{t \to 1} k$$ is equal to :

JEE Main 2022 (Online) 25th June Morning Shift
44

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

JEE Main 2022 (Online) 24th June Evening Shift
45

Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________.

JEE Main 2022 (Online) 24th June Morning Shift
46
Consider the parabola with vertex $$\left( {{1 \over 2},{3 \over 4}} \right)$$ and the directrix $$y = {1 \over 2}$$. Let P be the point where the parabola meets the line $$x = - {1 \over 2}$$. If the normal to the parabola at P intersects the parabola again at the point Q, then (PQ)2 is equal to :
JEE Main 2021 (Online) 1st September Evening Shift
47
The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :
JEE Main 2021 (Online) 31st August Morning Shift
48
If two tangents drawn from a point P to the
parabola y2 = 16(x $$-$$ 3) are at right angles, then the locus of point P is :
JEE Main 2021 (Online) 27th August Evening Shift
49
A tangent and a normal are drawn at the point P(2, $$-$$4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to :
JEE Main 2021 (Online) 27th August Morning Shift
50
Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of $$\Delta$$SOR is equal to :
JEE Main 2021 (Online) 25th July Morning Shift
51
Let P be a variable point on the parabola $$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :
JEE Main 2021 (Online) 20th July Evening Shift
52
Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :
JEE Main 2021 (Online) 20th July Morning Shift
53
Let L be a tangent line to the parabola y2 = 4x $$-$$ 20 at (6, 2). If L is also a tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over b} = 1$$, then the value of b is equal to :
JEE Main 2021 (Online) 17th March Evening Shift
54
Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is :
JEE Main 2021 (Online) 16th March Evening Shift
55
If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater than :
JEE Main 2021 (Online) 16th March Morning Shift
56
The shortest distance between the line x $$-$$ y = 1 and the curve x2 = 2y is :
JEE Main 2021 (Online) 25th February Evening Shift
57
A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?
JEE Main 2021 (Online) 25th February Morning Shift
58
If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates of P are :
JEE Main 2021 (Online) 24th February Evening Shift
59
The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parabola, is another parabola whose directrix is :
JEE Main 2021 (Online) 24th February Morning Shift
60
The centre of the circle passing through the point (0, 1) and touching the parabola
y = x2 at the point (2, 4) is :
JEE Main 2020 (Online) 6th September Evening Slot
61
Let L1 be a tangent to the parabola y2 = 4(x + 1)
and L2 be a tangent to the parabola y2 = 8(x + 2)
such that L1 and L2 intersect at right angles. Then L1 and L2 meet on the straight line :
JEE Main 2020 (Online) 6th September Morning Slot
62
If the common tangent to the parabolas,
y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2,
then c is equal to :
JEE Main 2020 (Online) 5th September Morning Slot
63
Let the latus ractum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2$$\sqrt 5 $$. Then, the distance between the centres of the circles C1 and C2 is :
JEE Main 2020 (Online) 3rd September Evening Slot
64
Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is $${4 \over 3}$$, then :
JEE Main 2020 (Online) 3rd September Morning Slot
65
The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola, is :
JEE Main 2020 (Online) 2nd September Evening Slot
66
If one end of a focal chord AB of the parabola y2 = 8x is at $$A\left( {{1 \over 2}, - 2} \right)$$, then the equation of the tangent to it at B is :
JEE Main 2020 (Online) 9th January Evening Slot
67
The locus of a point which divides the line segment joining the point (0, –1) and a point on the parabola, x2 = 4y, internally in the ratio 1 : 2, is :
JEE Main 2020 (Online) 8th January Morning Slot
68
If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :
JEE Main 2020 (Online) 7th January Morning Slot
69
The equation of common tangent to the curves y2 = 16x and xy = –4, is :
JEE Main 2019 (Online) 12th April Evening Slot
70
The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at the point :
JEE Main 2019 (Online) 12th April Evening Slot
71
Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the hyperbola 8x2 – y2 = 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :
JEE Main 2019 (Online) 12th April Morning Slot
72
If the line ax + y = c, touches both the curves x2 + y2 = 1 and y2 = 4$$\sqrt 2 $$x , then |c| is equal to :
JEE Main 2019 (Online) 10th April Evening Slot
73
The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2) and the x-axis is :-
JEE Main 2019 (Online) 9th April Evening Slot
74
If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is :
JEE Main 2019 (Online) 9th April Morning Slot
75
The tangent to the parabola y2 = 4x at the point where it intersects the circle x2 + y2 = 5 in the first quadrant, passes through the point :
JEE Main 2019 (Online) 8th April Evening Slot
76
The shortest distance between the line y = x and the curve y2 = x – 2 is :
JEE Main 2019 (Online) 8th April Morning Slot
77
The equation of a tangent to the parabola, x2 = 8y, which makes an angle $$\theta $$ with the positive directions of x-axis, is :
JEE Main 2019 (Online) 12th January Evening Slot
78
The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is :
JEE Main 2019 (Online) 12th January Morning Slot
79
Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $$\Delta $$PXQ is maximum. Then this maximum area (in sq. units) is :
JEE Main 2019 (Online) 12th January Morning Slot
80
If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :
JEE Main 2019 (Online) 11th January Evening Slot
81
The length of the chord of the parabola x2 $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$  is -
JEE Main 2019 (Online) 10th January Evening Slot
82
If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?
JEE Main 2019 (Online) 10th January Morning Slot
83
Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of $$\Delta $$ACB is maximum. Then, the area (in sq. units) of $$\Delta $$ACB, is :
JEE Main 2019 (Online) 9th January Evening Slot
84
Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?
JEE Main 2019 (Online) 9th January Morning Slot
85
Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :
JEE Main 2019 (Online) 9th January Morning Slot
86
If $$\theta $$ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan $$\theta $$| is equal to :
JEE Main 2019 (Online) 9th January Morning Slot
87
Let P be a point on the parabola, x2 = 4y. If the distance of P from the center of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is :
JEE Main 2018 (Online) 16th April Morning Slot
88
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and $$\angle $$CPB = $$\theta $$, then a value of tan$$\theta $$ is :
JEE Main 2018 (Offline)
89
Tangents drawn from the point ($$-$$8, 0) to the parabola y2 = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to :
JEE Main 2018 (Online) 15th April Evening Slot
90
Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $$3$$, then the equation of the common tangent to the two parabolas is :
JEE Main 2018 (Online) 15th April Morning Slot
91
If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $$\left| c \right|$$ is equal to :
JEE Main 2017 (Online) 9th April Morning Slot
92
If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is :
JEE Main 2017 (Online) 8th April Morning Slot
93
P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P passes through Q, then the minimum value of $$t_1^2$$ is :
JEE Main 2016 (Online) 10th April Morning Slot
94
Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the circle, $${x^2} + {\left( {y + 6} \right)^2} = 1$$. Then the equation of the circle, passing through $$C$$ and having its centre at $$P$$ is:
JEE Main 2016 (Offline)
95
Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the ratio $$1:3$$, then locus of $$P$$ is :
JEE Main 2015 (Offline)
96
The slope of the line touching both the parabolas $${y^2} = 4x$$ and $${x^2} = - 32y$$ is
JEE Main 2014 (Offline)
97
Given : A circle, $$2{x^2} + 2{y^2} = 5$$ and a parabola, $${y^2} = 4\sqrt 5 x$$.
Statement-1 : An equation of a common tangent to these curves is $$y = x + \sqrt 5 $$.

Statement-2 : If the line, $$y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right)$$ is their common tangent, then $$m$$ satiesfies $${m^4} - 3{m^2} + 2 = 0$$.

JEE Main 2013 (Offline)
98
If two tangents drawn from a point $$P$$ to the parabola $${y^2} = 4x$$ are at right angles, then the locus of $$P$$ is
AIEEE 2010
99
A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at :
AIEEE 2008
100
The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is :
AIEEE 2007
101
The locus of the vertices of the family of parabolas
$$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$ is :
AIEEE 2006
102
Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is :
AIEEE 2005
103
If $$a \ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$, then :
AIEEE 2004
104
The normal at the point$$\left( {bt_1^2,2b{t_1}} \right)$$ on a parabola meets the parabola again in the point $$\left( {bt_2^2,2b{t_2}} \right)$$, then :
AIEEE 2003
105
Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are :
AIEEE 2002
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